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The concatenation <math>L_1 \cdot L_2</math> of the formal languages <math>L_1\!</math> and <math>L_2\!</math> is just the cartesian product of sets <math>L_1 \times L_2</math> without the extra <math>\times</math>'s, but the relation of cartesian products to set-theoretic intersections and thus to logical conjunctions is far from being clear. One way of seeing a type of relation is to focus on the information that is needed to specify each construction, and thus to reflect on the signs that are used to carry this information. As a first approach to the topic of information, according to a strategy that seeks to be as elementary and as informal as possible, I introduce the following set of ideas, intended to be taken in a very provisional way.
The concatenation <math>\mathfrak{L}_1 \cdot \mathfrak{L}_2</math> of the formal languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> is just the cartesian product of sets <math>\mathfrak{L}_1 \times \mathfrak{L}_2</math> without the extra <math>\times</math>'s, but the relation of cartesian products to set-theoretic intersections and thus to logical conjunctions is far from being clear. One way of seeing a type of relation is to focus on the information that is needed to specify each construction, and thus to reflect on the signs that are used to carry this information. As a first approach to the topic of information, according to a strategy that seeks to be as elementary and as informal as possible, I introduce the following set of ideas, intended to be taken in a very provisional way.
A ''stricture'' is a specification of a certain set in a certain place, relative to a number of other sets, yet to be specified. It is assumed that one knows enough to tell if two strictures are equivalent as pieces of information, but any more determinate indications, like names for the places that are mentioned in the stricture, or bounds on the number of places that are involved, are regarded as being extraneous impositions, outside the proper concern of the definition, no matter how convenient they are found to be for a particular discussion. As a schematic form of illustration, a stricture can be pictured in the following shape:
A ''stricture'' is a specification of a certain set in a certain place, relative to a number of other sets, yet to be specified. It is assumed that one knows enough to tell if two strictures are equivalent as pieces of information, but any more determinate indications, like names for the places that are mentioned in the stricture, or bounds on the number of places that are involved, are regarded as being extraneous impositions, outside the proper concern of the definition, no matter how convenient they are found to be for a particular discussion. As a schematic form of illustration, a stricture can be pictured in the following shape:
# <math>p_1\!</math> says that <math>p\!</math> is in the first place of the product element under construction.
# <math>p_1\!</math> says that <math>p\!</math> is in the first place of the product element under construction.
# <math>q_2\!</math> says that <math>q\!</math> is in the second place of the product element under construction.
# <math>q_2\!</math> says that <math>q\!</math> is in the second place of the product element under construction.
Notice that, in construing the cartesian product of the sets <math>P\!</math> and <math>Q\!</math> or the concatenation of the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> in this way, one shifts the level of the active construction from the tupling of the elements in P and Q or the concatenation of the strings that are internal to the languages <math>\mathfrak{L}_1</math> and <math>\mathfrak{L}_2</math> to the concatenation of the external signs that it takes to indicate these sets or these languages, in other words, passing to a conjunction of indexed propositions, <math>P_1\!</math> and <math>Q_2,\!</math> or to a conjunction of assertions, <math>(\mathfrak{L}_1)_{(1)}</math> and <math>(\mathfrak{L}_2)_{(2)},</math> that marks the sets or the languages in question for insertion in the indicated places of a product set or a product language, respectively. In effect, the subscripting by the indices <math>^{\backprime\backprime} (1) ^{\prime\prime}</math> and <math>^{\backprime\backprime} (2) ^{\prime\prime}</math> can be recognized as a special case of concatenation, albeit through the posting of editorial remarks from an external ''mark-up'' language.
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In order to systematize the relations that strictures and straits placed
In order to systematize the relations that strictures and straits placed
at higher levels of complexity, constraint, information, and organization
at higher levels of complexity, constraint, information, and organization
